Academic literature on the topic 'Donaldson's theorem'

This paper contains two remarks about the application of the [Formula: see text]-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192(3) (2013) 717–750] concerning Conway mutation of alternating links.

AbstractWe prove that the Calabi–Yau equation can be solved on the Kodaira–Thurston manifold for all given T2-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic 4-manifolds with compatible but non-integrable almost complex structures.

Abstract We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

AbstractA theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact Kähler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.

AbstractIn [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]], the authors, in particular, associate to each finite quiver Q with a set of vertices I the so-called cohomological Hall algebra ℋ, which is ℤI≥0-graded. Its graded component ℋγ is defined as cohomology of the Artin moduli stack of representations with dimension vector γ. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can refine the grading to ℤI≥0×ℤ, and modify the product by a sign to get a super-commutative algebra (ℋ,⋆) (with parity induced by the ℤ-grading). It is conjectured in [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]] that in this case the algebra (ℋ⊗ℚ,⋆) is free super-commutative generated by a ℤI≥0×ℤ-graded vector space of the form V =Vprim ⊗ℚ[x] , where x is a variable of bidegree (0,2)∈ℤI≥0×ℤ, and all the spaces ⨁ k∈ℤVprimγ,k, γ∈ℤI≥0. are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs (γ,k) for which Vprimγ,k≠0 (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson–Thomas invariants, which was used by Mozgovoy to prove Kac’s conjecture for quivers with sufficiently many loops [S. Mozgovoy, Motivic Donaldson–Thomas invariants and Kac conjecture, Preprint (2011), arXiv:1103.2100v2[math.AG]]. Finally, we mention a connection with the paper of Reineke [M. Reineke, Degenerate cohomological Hall algebra and quantized Donaldson–Thomas invariants for m-loop quivers, Preprint (2011), arXiv:1102.3978v1[math.RT]].

AbstractThanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom–Sebastiani theorem in the case of regular functions. Moreover, slightly extending Hrushovski–Loeser’s construction adjusted to Sebag, Loeser and Nicaise’s motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions. The latter is meaningful as it has been a crucial element of constructing Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas invariants.

This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree k < 0, in the sense of [142]. We use this to show that the classical scheme X = t₀(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented d-critical locus, we prove in [18] that there is a natural perverse sheaf P·_X,s on X, and in [25], we construct a natural motive MF_X,s, in a certain quotient ring M^μ_X of the μ-equivariant motivic Grothendieck ring M^μ_X, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for k-shifted symplectic derived Artin stacks. We apply this theory to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections L??M of oriented Lagrangians L,M in an algebraic symplectic manifold (S,ω). In [23] we show that if (S,ω) is a complex symplectic manifold, and L,M are complex Lagrangians in S, then the intersection X= L??M, as a complex analytic subspace of S, extends naturally to a complex analytic d-critical locus (X, s) in the sense of Joyce [87]. If the canonical bundles K_L,K_M have square roots K^1/2_L, K^1/2_M then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·_L,M on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized Donaldson-Thomas invariants DT^α(τ) defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields K of characteristic zero, rather than K = C, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.

This thesis consists of the manuscripts of two research papers. In the first paper, we verify a recent conjecture of Kenyon/Szendroi by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁‚x₂‚‚ - x₃‚ƒx₄‚„ = 0}⊂ C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino (or dimer) shuffling algorithm of Elkies, Kuperberg, Larsen and Propp. In the second paper, we derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C³/G]. We need only the vertex operator methods of Okounkov-Reshetikhin-Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z₂‚‚ x Z₂‚‚, we will also use a refinement of the author's recent q-enumeration of pyramid partitions. In the appendix, written by Jim Bryan, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C³/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.

We use toric symmetry and blowups to study relationships in the Gromov-Witten theories of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. These two spaces are birationally equivalent via the common blowup space, the permutohedral variety. We prove an equivalence of certain invariants on blowups at only points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$ by showing that these invariants descend from the blowup. Further, the permutohedral variety has nontrivial automorphisms of its cohomology coming from toric symmetry. These symmetries can be forced to descend to the blowups at just points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. Enumerative consequences are discussed.

令X 為個帶有凱勒形式(Kähler form ω) 以及全純四形式( holomorphic four- form Ω )的四維緊致卡拉比丘空間(Calabi-Yau manifolds) 。在一些假設條件下，通過研究Donaldson- Thomas方程所決定的模空間，我們定義了四維Donaldson-Thomas不變量。我們也對四維局部卡拉比丘空間(local Calabi-Yau four-folds) 定義了四維Donaldson-Thomas 不變量，並且將之聯繫到三維Fano空間的Donaldson- Thomas 不變量。在一些情況下，我們還研究了DT/GW不變量對應。最后，我們在模空間光滑時計算了一些四維Donaldson- Thomas不變量。
Let X be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kahler form ω and a holomorphic four-form Ω. Under certain assumptions, we de ne Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifold. We also study sheaves counting on local Calabi-Yau four-folds. We relate the sheaves countings over X = KY with the Donaldson- Thomas invariants for the associated compact three-fold Y . In some specialcases, we prove the DT/GW correspondence for X. Finally, we compute the Donaldson-Thomas invariants of certain Calabi-Yau four-folds when the moduli spaces are smooth.
Detailed summary in vernacular field only.
Cao, Yalong.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.
Includes bibliographical references (leaves 100-105).
Abstracts also in Chinese.
Chapter 1 --- Introduction --- p.6
Chapter 2 --- The *4 operator --- p.18
Chapter 2.1 --- The *4 operator for bundles --- p.18
Chapter 2.2 --- The *4 operator for general coherent sheaves --- p.20
Chapter 3 --- Local Kuranishi structure of DT₄ moduli spaces --- p.22
Chapter 4 --- Compactification of DT₄ moduli spaces --- p.34
Chapter 4.1 --- Stable bundles compactification of DT₄ moduli spaces --- p.34
Chapter 4.2 --- Attempted general compactification of DT₄ moduli spaces --- p.36
Chapter 5 --- Virtual cycle construction --- p.39
Chapter 5.1 --- Virtual cycle construction for DT₄ moduli spaces --- p.40
Chapter 5.2 --- Virtual cycle construction for generalized DT₄ moduli spaces --- p.48
Chapter 6 --- DT4 invariants for compactly supported sheaves on local CY₄ --- p.52
Chapter 6.1 --- The case of X = KY --- p.52
Chapter 6.2 --- The case of X = T*S --- p.57
Chapter 7 --- DT₄ invariants on toric CY₄ via localization --- p.66
Chapter 8 --- Computational examples --- p.70
Chapter 8.1 --- DT₄=GW correspondence in some special cases --- p.71
Chapter 8.1.1 --- The case of Hol(X) = SU(4) --- p.72
Chapter 8.1.2 --- The case of Hol(X) = Sp(2) --- p.77
Chapter 8.2 --- Some remarks on cosection localizations for hyper-kähler four-folds --- p.79
Chapter 8.3 --- Li-Qin's examples --- p.80
Chapter 8.4 --- Moduli space of ideal sheaves of one point --- p.83
Chapter 9 --- Appendix --- p.85
Chapter 9.1 --- Local Kuranishi models of Mc° --- p.85
Chapter 9.2 --- Some remarks on the orientability of the determinant line bundles on the (generalized) DT₄ moduli spaces --- p.87
Chapter 9.3 --- Seidel-Thomas twists --- p.90
Chapter 9.4 --- A quiver representation of Mc --- p.92

In this thesis, we study the Donaldson-Thomas theory of local curves. The motivation is the Gromov-Witten/Donaldson-Thomas correspondence. First, we review the gauge theory motivation of the original construction and the history of the Donaldson-Thomas theory. Then we review the construction of Gieseker-Maruyama-Simpson moduli spaces and their relation with Hilbert schemes of threefolds. We also review the concept of a perfect obstruction theory and its relation with the virtual fundamental classes. Then we describe the Gromov-Witten/Donaldson-Thomas correspondence and the equivariant generalization. We study the equivariant Donaldson-Thomas theory of two types of threefolds. First, we consider the total space of P²-bundles over smooth curves of genus g with (C*)³ action along the fiber, and we consider the curve class of a section plus the class of a line in the fiber. We compute the Donaldson-Thomas partition function up to the first order and we find that it agrees with the Gromov-Witten prediction. The second case we consider is the local curve case. The threefold is the total space of a direct sum of two line bundles of opposite degree with an anti-diagonal action along the fiber. Based on the calculation from Gromov-Witten theory, the only non-trivial contributions to the Donaldson-Thomas partition function come from components that parametrize subschemes of pure dimension. We compute these contributions and find that they agree with the Gromov-Witten prediction. We are also able to verify the vanishing of contributions from the so-called product-type components.
Science, Faculty of
Mathematics, Department of
Graduate

The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.

This final chapter explores the range of ideas current in the contemporary animal ethics debate. Much of the chapter is devoted to documenting the critique of the animal welfare ethic, which holds that, while animals have moral standing, humans, being persons, have a superior moral status. Three different strands of this critique—based on utilitarian, rights, and contractarian approaches—are identified and explored. The final part of the chapter documents the fragmentation of the animal ethics debate in recent years. This has included a more nuanced position which seeks to decouple animal rights from abolitionism, accounts of animal ethics from virtue ethics and capabilities perspectives, and a relational turn associated with the feminist care ethic tradition and, more recently, the utilization of citizenship theory by Donaldson and Kymlicka.

Amid a growing ‘turn’ towards Southern cities, South African urban geographers continue to remind us why and how to attend to local context and draw on theory from elsewhere. Human Geographies of Stellenbosch: Transforming Space, Preserving Place? (edited by Ronnie Donaldson) provides a deep look at crucial questions facing one of South Africa’s most well-known town-cities. Written from years of local knowledge by scholars at Stellenbosch University, this volume asks what urban transformation means, who it is for, and the politically tantalising question of whether and how we might hold on to some of the old while aspiring towards the new? In a global context in which we are all searching for how to justly remember our messy past, how to decolonise and hold onto what makes places unique, this volume will be of interest to scholars asking such questions in and beyond urban studies.

Amid a growing ‘turn’ towards Southern cities, South African urban geographers continue to remind us why and how to attend to local context and draw on theory from elsewhere. Human Geographies of Stellenbosch: Transforming Space, Preserving Place? (edited by Ronnie Donaldson) provides a deep look at crucial questions facing one of South Africa’s most well-known town-cities. Written from years of local knowledge by scholars at Stellenbosch University, this volume asks what urban transformation means, who it is for, and the politically tantalising question of whether and how we might hold on to some of the old while aspiring towards the new? In a global context in which we are all searching for how to justly remember our messy past, how to decolonise and hold onto what makes places unique, this volume will be of interest to scholars asking such questions in and beyond urban studies.

AbstractStephen Clark’s article The Rights of Wild Things from 1979 was the starting point for the consideration in the animal ethics literature of the so-called ‘predation problem’. Clark examines the response of David George Ritchie to Henry Stephens Salt, the first writer who has argued explicitly in favor of animal rights. Ritchie attempts to demonstrate—via reductio ad absurdum—that animals cannot have rights, because granting them rights would oblige us to protect prey animals against predators that wrongly violate their rights. This article navigates the reader through the debate sparked off by Clarke’s article, with as final destination what I consider to be the best way to deal with the predation problem. I will successively discuss arguments against the predation reductio from Singer’s utilitarian approach, Regan’s deontological approach, Nussbaum’s capability approach, and Donaldson and Kymlicka’s political theory of animal rights.

This chapter argues that the Jones–Witten invariants can be generalized for smooth, nonsingular vector fields with invariant probability measure on three-manifolds, thus giving rise to new invariants of dynamical systems. After a short survey of cohomological field theory for Yang–Mills fields, Donaldson–Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. The chapter studies the case of Kähler manifolds by using the Witten's consideration of the strong coupling dynamics of N = 1 supersymmetric Yang–Mills theories. The whole construction is performed by implementing the notion of higher-dimensional asymptotic cycles. In the process Seiberg–Witten invariants are also described within this context. Finally, the chapter gives an interpretation of the asymptotic observables of four-manifolds in the context of string theory with flows.

This chapter provides elaboration of stakeholder theory, commencing with four general perspectives on stakeholder theory as identified by Donaldson and Preston (1995). This is followed by a discussion of how CSR or corporate social responsibility has influenced thinking about stakeholders and forms an integral part of the normative perspective. Carroll’s (1993) popular CSR model has been adapted and modified for this book, providing a more integrated and relevant approach. Defining and classifying stakeholders is the third major topic covered, drawing first on generic stakeholder theory and commencing with a discussion of primary and secondary, active and passive stakeholders. Particularly attention is given to the framework provided by Mitchell, Agle and Wood (1997) that defines ‘stockholder salience’ as a combination of ‘legitimacy, power and urgency’. These terms are explored in detail. The chapter concludes with an examination of event and tourism stakeholders, including a diagram and research notes from the events and tourism literature.

Chapter 5 draws on recent work in the so-called “political turn” in animal ethics, most notably Sue Donaldson and Will Kymlicka’s political theory of non-human animal rights, to discuss how these concepts can guide relations between groups of non-human animals and human political communities. The author discusses their proposals for animal citizenship and sovereignty. In the second half of this chapter, the author explains problems with traditional interpretations of sovereignty that rely on claims made by the powerful to legitimize the territorial domination of others. In order to challenge human sovereignty, we should challenge human superiority on all levels, including existing political systems. However, existing institutions and systems also hold a promise for other animals, and, like citizenship, these concepts can bring into focus new forms of interacting with other animals and institutionalizing these relations. In the final section, the author turns to examples of new ways of relating to other animals that can function as beginnings for further reformulating laws and political practices.

Abstract Human’s motion and its mechanisms had become interesting in the last years, where the medecine’s field search for rehabilitation methods for handicapped persons. Other fields, like sport sciences, professional or military world, search to distinguish profiles and ways to train them with specific purposes. Besides, recent findings in neuroscience try to describe these mechanisms from an organic point of view. Until now, different researchs had given a model about control motor that describes how the union between the senses’s information allows adaptable movements. One of this sense is the proprioception, the sense which has a quite big factor in the orientation and position of the body, its members and joints. For this reason, research for new strategies to explore proprioception and improve the theories of human motion could be done by three different vias. At first, the sense is analysed in a case-study where three groups of persons are compared in a controlled enviroment with three experimental tasks. The subjects belong to each group by the kind of sport they do: sedentary, normal sportsmen (e.g. athletics, swimming) and martial sportmen (e.g. karate, judo). They are compared thinking about the following hypothesis: “Martial Sportmen have a better proprioception than of the other groups’s subjects: It could be due to the type of exercises they do in their sports as empirically, a contact sportsman shows significantly superior motor skills to the members of the other two groups. The second via are records from encephalogram (EEG) while the experimental tasks are doing. These records are analised a posteriori with a set of processing algorithms to extract characteristics about brain’s activity of the proprioception and motion control. 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